I want to simulate a system whose impulse response is like the following:
$$h(t) = e^{-at} \sin(t)$$
The graph of which should look similar to the plot below:
I want to simulate the output for this system in response to an arbitrary input signal. I believe I can use control theory to do this. Is the impulse response sufficient to derive a state space model, and how do I do it? Under what conditions is the impulse response sufficient?